7:00 - Wait a minute! You are drawing a false conclusion from the test. You can't conclude that example b is convergent by the nth term test simply because the limit of the sequence a_n = 0; that's not how the test works. When the limit of a_n = 0, this particular test is *inconclusive*, which is to say that the series MAY converge, but it may not as well. You can only establish the convergence/divergence of this series by using another test, such as the Integral Test or the Limit Comparison Test. The nth term test has nothing to say about convergence of series; it can only conclusively establish divergence. Recall that there are sequences whose limit = 0, but whose series are in fact divergent (e.g. the harmonic series), it is not possible to generalize the result in way that you have done here. Furthermore, the limit of the series in example b is in fact divergent, not convergent.
In the three examples for the nth term test, each have different starting points for n (n=1 or n=0), however the test states that it starts at n=1, therefore for the first example, wouldnt you have to rework it to get to n=1? It will still converge but I wanted to make sure whether or not rewriting a(n) is applicable to ALL test and not SOME test
7:00 - Wait a minute! You are drawing a false conclusion from the test. You can't conclude that example b is convergent by the nth term test simply because the limit of the sequence a_n = 0; that's not how the test works. When the limit of a_n = 0, this particular test is *inconclusive*, which is to say that the series MAY converge, but it may not as well. You can only establish the convergence/divergence of this series by using another test, such as the Integral Test or the Limit Comparison Test. The nth term test has nothing to say about convergence of series; it can only conclusively establish divergence. Recall that there are sequences whose limit = 0, but whose series are in fact divergent (e.g. the harmonic series), it is not possible to generalize the result in way that you have done here. Furthermore, the limit of the series in example b is in fact divergent, not convergent.
Yea you are indeed correct
In the three examples for the nth term test, each have different starting points for n (n=1 or n=0), however the test states that it starts at n=1, therefore for the first example, wouldnt you have to rework it to get to n=1? It will still converge but I wanted to make sure whether or not rewriting a(n) is applicable to ALL test and not SOME test
As other comments are explained, nth term test is only ensures diverge, it is inconculsive for convergence.
Your video are very helpful mam...I saw your full playlist of Discrete Mathematics... 👍👍👍
Thanks a lot
With all due respect, this video is incorrect, especially as the other comment pointed out.
Thanks. I'll take a look.